12. If two circles are concentric, show that the area of the ring between their circumferences is equal to the area of a circle having for its diameter a chord of the larger circle that is tangent to the smaller.

13. Find the area of the sector of a circle intercepting an arc of 50°, the radius of the circle being 10 ft. [§ 776.]

14. The radius of a circle is 20 ft. What is the angle of a sector having an area of 300 sq. ft. ?

15. The radius of a circle is 20 ft., and the area of a sector of the circle is 300 sq. ft. Find the area of a similar sector in a circle having

a radius 50 ft. long.

16. What is the radius of a circle having an area equal to 16 times the area of a circle with a radius 5 ft. long?

17. Find the area of a circle circumscribed about a square having an area of 600 sq. ft. [§ 729.]

18. Show that the area of a circumscribed equilateral triangle is greater than that of a square circumscribed about the same circle.

19. Four circles, each with a radius 5 ft. long, have their centers at the vertices of a square, and are tangent. Find the area of a circle tangent to all of them.

20. How many degrees in the arc, the length of which is equal to the radius of the circle?

21. A circle is circumscribed about the rightangled triangle ABC. Semicircles are described on the two legs as diameters. Prove that the sum of the crescents ADBE and BFCG is equivalent to the triangle ABC.

22. The radius of a regular inscribed polygon is a mean proportional between its apothem and the radius of a similar circumscribed polygon.

23. If the bisectors of the angles of a polygon meet in a point, a circle can be inscribed in the polygon.

24. The diagonals of a regular pentagon form B by their intersection a second regular pentagon.

25. Any two diagonals of a regular pentagon not drawn from a common vertex divide each other into extreme and mean ratio. [A ABC and CfD are similar.]

D

E

26. Divide an angle of an equilateral triangle into five equal parts.

27. If two angles at the centers of unequal circles are subtended by arcs of equal length, the angles are inversely proportional to the radii of the circles.

28. The apothem of a regular inscribed pentagon is equal to one half the sum of the radius of the circle and a side of a regular inscribed decagon.

29. If two chords of a given circle intersect each other at right angles, and on the four segments of the chords as diameters, circles are de scribed, the sum of the four circles is equivalent to the given circle. [Ex. 34, page 217.]

30. Divide a circle into three equivalent parts by concentric circles (§ 784).

31. The radius of a given circle ABD is 10 ft. Find the areas of the two segments BCA and BDA into which the circle is divided by a chord AB equal in length to the radius. [Subtract area of ▲ from area of sector.]

32. Find the radius of a circle that is doubled in area by increasing its radius one foot.

33. On the sides of a square as diameters, four semicircles are described within the square, forming four leaves. If the side of the square is α, find the area of the leaves.

34. In a given equilateral triangle inscribe three equal circles tangent to each other and to the sides of the triangle.

35. In a given circle inscribe three equal circles tangent to each other and to the given circle.

36. In the circle ABCD, the diameters AC and BD are at right angles to each other. With A E, the middle point of OC, as a center, and EB as a radius, the arc BF is described. Prove that the radius OA is divided into extreme and mean ratio at F.

[Describe arc OG with E as center, and arc GH with B as center.]

C

A

B

A

H

E

37. If a regular polygon of n sides be circumscribed about a circle, the sum of the perpendiculars from the points of contact to any tangent to the circle is equal to n times the radius.

[If A, B, C, D, etc., are the points of contact of the polygon and P the point at which the tangent is drawn, the sum of the Is from A, B, etc., on tangent at P = sum of is from P to tangents drawn at A, B, etc.; and this by Ex. 8= nR.]

38. The sum of the perpendiculars from the vertices of a regular inscribed polygon to any line without the circle is equal to n times the perpendicular from the center of the circle to the line.

[Draw a tangent to the O parallel to the given line, and then use Ex. 39.] 39. The sum of the squares of the lines drawn from any point in the circumference to the vertices of a regular inscribed polygon is equal to 2 nR2. [Using notation of Ex. 39, show that the square

of the line from the given point P to each vertex =2 R times the L from the vertex to a tangent at P. Add these equations and use Ex. 39.]

40. A crescent-shaped region is bounded by a semi-circumference of radius a, and another circular arc whose center lies on the semi-circumference produced. Find the area and the perimeter of the region.

[Show that the arc is a quadrant in a O with radius = a √2.]

41. Three points divide a circumference into equal parts. Through each pair of these points

an arc of a circle is described tangent to the

radii drawn to the points and lying wholly within the circle.

Find

the perimeter of the figure thus formed, and show that its area is

3 (√3 − 1π) α2, where a denotes the radius of the circle. [Show that each arc is of a circumference

with radius a √3.]

42. Three radii are drawn in a circle of radius 2 a, so as to divide the circumference into three equal parts; and, with the middle of these radii as centers, arcs are drawn, each with the radius a, so as to form a closed figure

(trefoil). Show that the length of the perimeter of the trefoil is equal to that of the circle, and find its area.

SOLID GEOMETRY

INDEX OF MATHEMATICAL TERMS

SOLID GEOMETRY

[The references are to articles.]